Transition probabilities of a continuous time Markov chain at time t

Computes the transition matrix \(P(t) = \exp(t Q)\) of a continuous time Markov chain with generator \(Q\), or one of its rows.

Usage

probabilityatT(C, t, x0, useRCpp = TRUE)

Arguments

C

A ctmc object.

t

Time, a non-negative number.

x0

Optional index (position in states(C)) of the initial state.

useRCpp

Logical: compute the matrix exponential in C++ (default) or with expm::expm()?

Value

Without x0, the matrix \(P(t)\), with the states as dimnames, whose entry \((i, j)\) is the probability of being in state j at time t when starting from i. With x0, row x0 of that matrix: the distribution of the state at time t when starting from x0.

References

Dobrow, R. P. (2016). Introduction to Stochastic Processes with R. Wiley.

See also

Other continuous time Markov chains: ExpectedTime(), ctmcFit(), freq2Generator(), generatorToTransitionMatrix(), impreciseProbabilityatT(), is.CTMCirreducible(), is.TimeReversible(), rctmc(), transition2Generator()

Author

Vandit Jain

Examples

states <- c("a", "b", "c", "d")
gen <- matrix(c(-1, 1/2, 1/2, 0,
                1/4, -1/2, 0, 1/4,
                1/6, 0, -1/3, 1/6,
                0, 0, 0, 0), nrow = 4, byrow = TRUE,
              dimnames = list(states, states))
ctmc <- new("ctmc", states = states, byrow = TRUE, generator = gen,
            name = "testctmc")
probabilityatT(ctmc, 1)
#>            a          b         c          d
#> a 0.41546882 0.24714119 0.2703605 0.06702946
#> b 0.12357060 0.63939068 0.0348290 0.20220972
#> c 0.09012017 0.02321933 0.7411205 0.14553997
#> d 0.00000000 0.00000000 0.0000000 1.00000000
probabilityatT(ctmc, 1, x0 = 1)
#> [1] 0.41546882 0.24714119 0.27036052 0.06702946